Lemma 5.24.3D1DAB1D38637

Proof. On a coordinate ball, add the same smooth function $q$ to both functions so that $a+q$ and $b+q$ are plurisubharmonic. By the submean inequality and upper semicontinuity, a plurisubharmonic function is, at each point, the limit of its averages over balls centered there as the radius tends to zero. The almost-everywhere inequality orders these averages, so passing to the limit gives $a+q\leq b+q$. Subtracting $q$ proves the claim. $\square$